| L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 4·7-s + 8-s + 9-s + 2·10-s − 11-s + 12-s − 2·13-s + 4·14-s + 2·15-s + 16-s + 17-s + 18-s − 4·19-s + 2·20-s + 4·21-s − 22-s − 4·23-s + 24-s − 25-s − 2·26-s + 27-s + 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s − 0.554·13-s + 1.06·14-s + 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.872·21-s − 0.213·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.883262136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.883262136\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.02551625906787275101266302661, −8.886846373788988337486830690127, −8.133863543051193292800465506537, −7.40331969331034486175214092642, −6.35174311331749427198224792212, −5.35008796069879898888153935324, −4.75423241604491376136715566548, −3.68721331750950772544404465331, −2.28830201610711163759807511530, −1.75881684451212862226705712506,
1.75881684451212862226705712506, 2.28830201610711163759807511530, 3.68721331750950772544404465331, 4.75423241604491376136715566548, 5.35008796069879898888153935324, 6.35174311331749427198224792212, 7.40331969331034486175214092642, 8.133863543051193292800465506537, 8.886846373788988337486830690127, 10.02551625906787275101266302661
